On slim double Lie groupoids (Q433528)

In a previous paper [J. Pure Appl. Algebra 213, No. 6, 1031--1045 (2009; Zbl 1222.18004)], \textit{N. Andruskiewitsch} and \textit{S. Natale} have proved that the category of slim double groupoids , with fixed vertical and horizontal groupoids \({\mathcal V}\) and \({\mathcal H}\), satisfying the filling condition, is equivalent to the category of diagrams over \({\mathcal V}\) and \({\mathcal H}\).NEWLINENEWLINE In the carefully written paper under review, the authors extend the result just mentioned to the context of double Lie groupoids. More precisely, they prove that the category of slim double Lie groupoids with fixed vertical and horizontal Lie groupoids \({\mathcal V}\) and \({\mathcal H}\), and proper core action, is equivalent to the category of diagrams of Lie groupoids \(({\mathcal D},j,i)\) such that the maps \(j\) and \(i\) are transversal at the identities.